If Purn is 6 years older than Sum and Sharvn is twice as old as Purn, work out how old are Sum and Sharvn if the average of their ages is 54?
Answer (2)
Define variables: S = Sum's age, P = Purn's age, T = Sturon's age.
Express Purn's and Sturon's ages in terms of Sum's age: P = S + 6 , T = 2 ( S + 6 ) .
Substitute into the average equation and solve for S : 3 S + ( S + 6 ) + ( 2 S + 12 ) = 54 , which gives S = 36 .
Find Purn's and Sturon's ages: P = 42 , T = 84 . The ages are S = 36 , P = 42 , T = 84 .
Explanation
Understanding the Problem Let's break down this age problem step by step! We're given that Purn is older than Sum, Sturon is older than Purn, and we know the average of their ages. Our goal is to find each of their ages. Let's use algebra to solve this.
Setting up Equations Let's use variables to represent each person's age:
S = Sum's age P = Purn's age T = Sturon's age
Now, let's translate the given information into equations:
Purn is 6 years older than Sum: P = S + 6 Sturon is twice as old as Purn: T = 2 P The average of their ages is 54: 3 S + P + T = 54
Expressing Ages in Terms of S We can use substitution to solve this system of equations. First, let's substitute P = S + 6 into the equation T = 2 P :
T = 2 ( S + 6 ) = 2 S + 12
Now we have both P and T in terms of S .
Solving for Sum's Age Next, substitute P = S + 6 and T = 2 S + 12 into the average equation:
3 S + ( S + 6 ) + ( 2 S + 12 ) = 54
Simplify the equation:
3 4 S + 18 = 54
Multiply both sides by 3:
4 S + 18 = 162
Subtract 18 from both sides:
4 S = 144
Divide both sides by 4:
S = 36
So, Sum is 36 years old.
Finding Purn's and Sturon's Ages Now that we know Sum's age, we can find Purn's and Sturon's ages:
Purn's age: P = S + 6 = 36 + 6 = 42 Sturon's age: T = 2 P = 2 ( 42 ) = 84
Therefore, Purn is 42 years old and Sturon is 84 years old.
Final Answer Let's summarize our findings:
Sum is 36 years old. Purn is 42 years old. Sturon is 84 years old.
Examples
Understanding how to solve age-related problems using algebra is useful in many real-life situations. For example, you might use these skills to determine future ages based on current relationships, or to solve problems involving family inheritance where ages and relationships are key factors. This type of algebra helps in planning and making predictions based on known information.
Sum is 36 years old, Purn is 42 years old, and Sharvn is 84 years old. The ages were derived through algebraic substitution based on the relationships given in the problem. The average of their ages confirms the solution as accurate.
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